Birkhoff

Garrett Birkhoff

1911-1996

In his article
"Subgroups of Abelian Groups"
(Proceedings London Math Soc 38 (1935))
he raises the problem

to determine
"...relative invariants of subgroups;
i.e. invariants under automorphisms"
of the given group.

Birkhoff's paper contains many partial results. For example:
For A of type (6,4,2), the number of isomorphism classes of pairs (A,B) tends to infinity with p. (Cor 15.1).
Actually, as we will show: for B of type (4,2), there are p+7 indecomposable pairs and 10 decomposable pairs.
His main method: To represent the embedding B → A by a matrix
(and often one can use as entries just p-powers).